Sauté the vegetables: In a large pot, melt the butter over medium heat. Add the diced onion, celery, and carrots. Cook until the vegetables are softened, about 5- 7 minutes.
Add cauliflower and broth: Stir in the chopped cauliflower and chicken broth. Bring the mixture to a boil, then reduce the heat and let it simmer for 10-15 minutes, or until the cauliflower is tender.
Add chicken and cream: Stir in the shredded chicken and heavy cream. Cook on low heat for another 5 minutes, allowing the flavors to meld together.
Season and serve: Season the soup with salt and pepper to taste. Garnish with fresh parsley before serving.